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New abstract Hardy spaces. (English) Zbl 1171.42012
Let $$\mathbb{X}$$ be a set, $$d$$ a quasi-distance on $$\mathbb{X}$$ and $$\mu$$ a Borel measure which satisfies the doubling property. For $$Q$$ a ball and $$i\geq0$$, set $$S_i(Q)=\{ x\in \mathbb{X }:2^i \leq 1+\frac {d(x,c(Q))}{r_Q}<2^{i+1}\}$$, where $$r_Q$$ is the radius of the ball $$Q$$ and $$c(Q)$$ its center. Let $$\mathcal L=\{B(x,r): x\in \mathbb{X}, r>0\}$$ and $$\mathcal B=(B_Q)_{Q\in \mathcal L}$$ be a collection of uniformly $$L^2$$-bounded linear operator, indexed by the collection $$\mathcal L$$. Let $$\varepsilon>0$$ be a fixed parameter. A function $$m\in L_{\text{loc}}^1$$ is called an $$\epsilon$$-molecule associated to a ball $$Q$$ if there exists a real function $$f_Q$$ such that $$m=B_Q(f_Q)$$ with for all $$i\geq0,$$ $$\|f_Q\|_{2,S_i(Q)}\leq[\mu(2^iQ)]^{-\frac{1}{2}}2^{-\varepsilon i}$$. If in addition $$\text{supp}(f_Q)\subset Q$$, then $$m=B_Q(f_Q)$$ is called an atom.
In this paper, the authors construct abstract Hardy spaces by a molecular (or atomic) decomposition and use the weakest assumptions to obtain good properties for these spaces. Mainly, the authors obtain a criterion for the continuity of an operator from the Hardy space into $$L^1(\mathbb X)$$ and an interpolation result between the Hardy space and Lebesgue spaces. The authors also obtain some results on weighted norm inequalities and present partial results in order to understand a characterization of the duals of Hardy spaces.

MSC:
 42B35 Function spaces arising in harmonic analysis 42B30 $$H^p$$-spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 43A99 Abstract harmonic analysis
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